Studying eigenvalues and eigenvectors of an adjacency matrix of a graph is one of the most important topics in algebraic graph theory; especially considering its many applications in combinatorics, chemistry and theoretical computer science [1]} and [2]}. Mixed graphs can be considered as a generalization of graphs. Further, they have more applications than graphs. Even though research about spectrum of mixed graphs is rare. One of the reasons for such rareness is that its adjacency matrix is not symmetric. Which means some of its eigenvalues are complex. Mohar defined an interesting general Hermitian adjacency matrix of mixed graphs as follows [3]} :
Let \(X\)  be a mixed graph with \(n\)  vertices and \(\alpha \)  be the unit complex number \(e^{i\theta }\)  . Then, the \(\alpha \) -Hermitian adjacency matrix of \(X\)  is an \(n\times n\)  matrix \(H_{\alpha }(X)=(h_{ij})\) , where
\(h_{ij} = \left\lbrace \begin{array}{ll}1 & \text{if } u_iv_j \in E(D),\\\alpha & \text{if } \vec{u_iv_j} \in \vec{E}(D), \\\overline{\alpha } & \text{if } \vec{v_ju_i} \in \vec{E}(D),\\0 & \text{otherwise}.\end{array}\right.\) 
